Question

How do you factor `y=2x^2 - 5x – 3`?

Answer 1

`y = (2x+1)(x-3)`

Explanation:

To find the factors, put `y=0`

`=>2x^2-5x-3=0`

This is a quadratic equation and will have 2 factors or 2 roots.

We first find two numbers that:

• multiply to `color(red)(-6)` (because `2xx-3=-6`) and
• add up to `color(red)(-5)`.

Let's list the factors of -6:

`color(blue)(1xx-6=-6)`
`2xx-3=-6`
`3xx-2=-6`
`6xx-1=-6`

From the above combinations, `1 + (-6) =1-6= -5`.

Now back to the quadratic equation:

`2x^2-5x-3=0`

Here, we write `-5` as `1-6`.
Then `-5x` will be `x-6x`

`=>2x^2color(red)(+1x-6x)-3=0`

Make 2 pairs:
`=>color(orange)(2x^2+x)color(blue)(-6x-3)=0`

Take out the common terms from each pair:

`=>color(orange)(x(2x+1))color(blue)(-3(2x+1))=0`

`color(red)(2x+1)` is common to the two terms in the equation. Take out the common terms, and write the remaining terms:

`=>color(red)((2x+1))color(green)((x-3))=0`

`y = (2x+1)(x-3)`

We can check our answer by working backwards:

i.e. solve: `y = (2x+1)(x-3)`

`y = 2x(x-3)+1(x-3)`

`y = 2x^2-6x+x-3`

`y = 2x^2-5x-3`